Discriminative and generative methods for bags of features

Zebra

Non-zebra

Many slides adapted from Fei-Fei Li, Rob Fergus, and Antonio Torralba

Image classification• Given the bag-of-features representations of

images from different classes, how do we learn a model for distinguishing them?

Discriminative methods• Learn a decision rule (classifier) assigning

bag-of-features representations of images to different classes

Zebra

Non-zebra

Decisionboundary

Classification• Assign input vector to one of two or more

classes• Any decision rule divides input space into

decision regions separated by decision boundaries

Nearest Neighbor Classifier

• Assign label of nearest training data point to each test data point

Voronoi partitioning of feature space for two-category 2D and 3D data

from Duda et al.

Source: D. Lowe

• For a new point, find the k closest points from training data• Labels of the k points “vote” to classify• Works well provided there is lots of data and the distance function

is good

K-Nearest Neighbors

k = 5

Source: D. Lowe

Functions for comparing histograms

• L1 distance

• χ2 distance

• Quadratic distance (cross-bin)

N

i

ihihhhD1

2121 |)()(|),(

Jan Puzicha, Yossi Rubner, Carlo Tomasi, Joachim M. Buhmann: Empirical Evaluation of Dissimilarity Measures for Color and Texture. ICCV 1999

N

i ihihihihhhD

1 21

221

21 )()()()(),(

ji

ij jhihAhhD,

22121 ))()((),(

Earth Mover’s Distance• Each image is represented by a signature S consisting

of a set of centers {mi } and weights {wi }• Centers can be codewords from universal vocabulary,

clusters of features in the image, or individual features (in which case quantization is not required)

• Earth Mover’s Distance has the form

where the flows fij are given by the solution of a transportation problem

Y. Rubner, C. Tomasi, and L. Guibas: A Metric for Distributions with Applications to Image Databases. ICCV 1998

ji ij

jiij

fmmdf

SSEMD,

2121

),(),(

Linear classifiers• Find linear function (hyperplane) to separate

positive and negative examples

0:negative0:positive

bb

ii

ii

wxxwxx

Which hyperplaneis best?

Support vector machines• Find hyperplane that maximizes the margin

between the positive and negative examples

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

Support vector machines• Find hyperplane that maximizes the margin

between the positive and negative examples

1:1)(negative1:1)( positive

byby

iii

iii

wxxwxx

MarginSupport vectors

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

Distance between point and hyperplane: ||||

||wwx bi

For support vectors, 1 bi wx

Therefore, the margin is 2 / ||w||

Finding the maximum margin hyperplane1. Maximize margin 2/||w||2. Correctly classify all training data:

Quadratic optimization problem:

Minimize

Subject to yi(w·xi+b) ≥ 1

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

wwT

21

1:1)(negative1:1)( positive

byby

iii

iii

wxxwxx

Finding the maximum margin hyperplane• Solution:

i iii y xw

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

Support vector

learnedweight

Finding the maximum margin hyperplane• Solution:

b = yi – w·xi for any support vector

• Classification function (decision boundary):

• Notice that it relies on an inner product between the test point x and the support vectors xi

• Solving the optimization problem also involves computing the inner products xi · xj between all pairs of training points

i iii y xw

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

bybi iii xxxw

• Datasets that are linearly separable work out great:

• But what if the dataset is just too hard?

• We can map it to a higher-dimensional space:

0 x

0 x

0 x

x2

Nonlinear SVMs

Slide credit: Andrew Moore

Φ: x → φ(x)

Nonlinear SVMs• General idea: the original input space can

always be mapped to some higher-dimensional feature space where the training set is separable:

Slide credit: Andrew Moore

Nonlinear SVMs• The kernel trick: instead of explicitly computing

the lifting transformation φ(x), define a kernel function K such that

K(xi , xj) = φ(xi ) · φ(xj)

(to be valid, the kernel function must satisfy Mercer’s condition)

• This gives a nonlinear decision boundary in the original feature space:

bKyi

iii ),( xx

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

Kernels for bags of features• Histogram intersection kernel:

• Generalized Gaussian kernel:

• D can be Euclidean distance, χ2 distance, Earth Mover’s Distance, etc.

N

i

ihihhhI1

2121 ))(),(min(),(

2

2121 ),(1exp),( hhDA

hhK

J. Zhang, M. Marszalek, S. Lazebnik, and C. Schmid, Local Features and Kernels for Classifcation of Texture and Object Categories: A Comprehensive Study, IJCV 2007

Summary: SVMs for image classification1. Pick an image representation (in our case, bag

of features)2. Pick a kernel function for that representation3. Compute the matrix of kernel values between

every pair of training examples4. Feed the kernel matrix into your favorite SVM

solver to obtain support vectors and weights5. At test time: compute kernel values for your test

example and each support vector, and combine them with the learned weights to get the value of the decision function

What about multi-class SVMs?• Unfortunately, there is no “definitive” multi-

class SVM formulation• In practice, we have to obtain a multi-class

SVM by combining multiple two-class SVMs • One vs. others

• Traning: learn an SVM for each class vs. the others• Testing: apply each SVM to test example and assign to it the

class of the SVM that returns the highest decision value

• One vs. one• Training: learn an SVM for each pair of classes• Testing: each learned SVM “votes” for a class to assign to

the test example

SVMs: Pros and cons• Pros

• Many publicly available SVM packages:http://www.kernel-machines.org/software

• Kernel-based framework is very powerful, flexible• SVMs work very well in practice, even with very small

training sample sizes

• Cons• No “direct” multi-class SVM, must combine two-class SVMs• Computation, memory

– During training time, must compute matrix of kernel values for every pair of examples

– Learning can take a very long time for large-scale problems

Summary: Discriminative methods• Nearest-neighbor and k-nearest-neighbor

classifiers• L1 distance, χ2 distance, quadratic distance,

Earth Mover’s Distance

• Support vector machines• Linear classifiers• Margin maximization• The kernel trick• Kernel functions: histogram intersection, generalized Gaussian,

pyramid match• Multi-class

• Of course, there are many other classifiers out there• Neural networks, boosting, decision trees, …